Bee skeps provide beeswax and honey, and also help crops grow faster in a radius of 13 tiles.
To collect beeswax, just right click the skep and harvest it.
To collect honey, left click a bucket and right click on the skep.

Working with Bee Skep randomly adds Beesting, each new wound would be worse by 3 HHP. There is a chance to get A Bee That Stung when wounded.

Pollination

Bee Skep will reduce the time it takes for crops to fully mature; with the exception of Barley and Wheat, which are not affected by bee skeps.

"Each time a crop grows to the next stage, it checks if there's a beehive within range. If it finds beehive, the time before the next stage change is reduced, and the beehive receives a usage tick. Beehives can store up to 50 uses (and noting how jorb worded it, it's likely they start will a full 50 when first created), and will regenerate 6 uses per hour. Crops using a beehive will cause honey and wax production (exact mechanics unknown), but jorb has stated there is a low level of base production of honey/wax by a hive even if it has no crops nearby." -Sevenless

Without any pollation by crops (ie inside houses or caves), Bee Skeps take less than 4 hours to produce some honey, and about 13 days to produce a piece of wax. These will be low quality.

General Notes

The bee skep production is affected by maturing plants in its vicinity. Therefore, if you want a lot of wax or honey, you have to actively farm around the bee skep. However, skeps cannot "share" this resource. Two skeps sitting side by side will produce half the honey/wax as a single skep in that location would.

It is still unclear how and when exactly a bee skep will produce wax and honey.

Quality

Quality of the Bee Skep is an average of all used materials, and soft-caps the quality of its products.

Quality of wax and honey produced seems to be affected by the quality of crops covered by the bee hive (All crops, except barley and wheat, will affect a beehive this way)